How the heck is this done

You could do a wiggle expression (in AE) in conjunction with some time remapping, but that may not give you the control you want because wiggle can be hard to make a specific look with. You could also try wiggle with a lens distortion effect, or scale. Lots of options. I have never used it myself, but the Twitch plug-in from VideoCopilot might be useful for this.
 
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